Question

Find $\frac{dy}{dx},x=a\left(\cos t+\log \tan \frac{t}{2}\right),y=a\sin t$

Answer 1

$x=a(\cos t+\log \tan \frac{t}{2})$ ; $y=a\sin t$

$\frac{dx}{dt}=a\left(-\sin t+\frac{1}{\tan t/2}\times {\sec }^{2}t/2\times \frac{1}{2}\right)$

$=a(-\sin t+\frac{1}{2}\frac{\cos t/2}{\sin t/2{\cos }^{2t/2}})$

$=a(-\sin t+\frac{1}{\sin t})$

$=a\left(\frac{{\cos }^{2}t}{\sin t}\right)$

$\frac{dy}{dx}=a\cos t$

$\therefore \frac{dy}{dx}=\frac{a\cos t}{a\left(\frac{\cos 2t}{\sin t}\right)}$

$=\frac{\sin t}{\cos t}=\tan t$